Which integral gives the length of the graph of $y=x\ln(x)$ from the point $(1, 0)$ to $(e, e)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $ \int_1^e\sqrt{\ln x+2}~dx$ (Choice B) B $ \int_1^e\sqrt{\ln^2x+2\ln x+2}~dx$ (Choice C) C $ \int_1^e\sqrt{\ln^2x+2\ln x+3}~dx$ (Choice D) D $ \int_1^e\sqrt{1+\ln^2x}~dx$
Solution: Recall that the formula for arc length of $~f(x)~$ over $~[a, b]~$ is $ L=\int_a^b\sqrt{1+\big[f\,^\prime(x)\big]^2}~dx\,$. First calculate $~f\,^\prime(x)$. $ f(x)=x\ln x\Rightarrow f\,^\prime(x)=\ln x+1$ Next, use the formula above to write the integral expression that gives the arc length in question and simplify. $ L=\int_{1}^e\sqrt{1+\Big[\ln x+1\Big]^2}~dx=\int_1^e\sqrt{\ln^2x+2\ln x+2}~dx$